To this aim, we use a quantitative version of the Chebotarev density theorem. Intuitively speaking, given a Galois extension \(L\)  of \(\mathbb {Q}\) , the theorem gives a bound on the number of primes splitting in a certain pattern in \(\mathcal {O}_L\) . The different classes of splitting patterns correspond to conjugacy classes of the Galois group \(\operatorname{Gal}(L/\mathbb {Q})\)  of \(L\) . As it turns out, the primes splitting completely in \(L\)  correspond to the conjugacy class \(\lbrace id\rbrace \) 
containing solely the identity element \(id\)  of \(\operatorname{Gal}(L/\mathbb {Q})\) . The asymptotic version of the theorem then asserts that the set of completely split primes has density \(\frac{1}{|\operatorname{Gal}(L/\mathbb {Q})|}\) . Denoting by \(\pi (x)\)  the number of all rational primes less or equal to \(x\) , and by \(\pi _1(x)\)  the number of completely split primes, the quantitative version of the theorem is as follows [1]}, [2]}:
